{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "# Shooting Method"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 7,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as pt"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 8,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "def rk4_step(y, t, h, f):\n",
        "    k1 = f(t, y)\n",
        "    k2 = f(t+h/2, y + h/2*k1)\n",
        "    k3 = f(t+h/2, y + h/2*k2)\n",
        "    k4 = f(t+h, y + h*k3)\n",
        "    return y + h/6*(k1 + 2*k2 + 2*k3 + k4)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Want to solve:\n",
        "\n",
        "$$w''(t)=\\frac 32w^2$$\n",
        "\n",
        "with $w(0)=4$ and $w(1)=1$. (Example due to Stoer and Bulirsch)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 9,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "def f(t, y):\n",
        "    w, w_prime = y\n",
        "    return np.array([w_prime, 3/2*w**2])"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "The following function carries out the shooting method for a given $w'(0)$ using RK4:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 20,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "def shoot(w_prime):\n",
        "    times = [0]\n",
        "    y_values = [np.array([4, w_prime])]\n",
        "    \n",
        "    h = 1/2**7\n",
        "    t_end = 1\n",
        "    \n",
        "    while times[-1] < t_end:\n",
        "        y_values.append(rk4_step(y_values[-1], times[-1], h, f))\n",
        "        times.append(times[-1]+h)\n",
        "    \n",
        "    y_values = np.array(y_values)\n",
        "    \n",
        "    # actually floating-point-equal due to power-of-2 h\n",
        "    assert times[-1] == t_end\n",
        "    \n",
        "    print(\"w'(0) = %g  ->  w(1)= %.5g\" % (w_prime, y_values[-1,0]))\n",
        "\n",
        "    pt.plot(times, y_values[:, 0], label=\"$w'(0)=%.2g$\" % w_prime)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Call `shoot` to see if you can solve the boundary value problem.\n",
        "\n",
        "Start with $w'(0)=0$.\n",
        "\n",
        "(You may call `pt.legend` to take advantage of automatic labeling.)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 19,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "w'(0) = 0  ->  w(1)= 87.08\n",
            "w'(0) = -5  ->  w(1)= 12.058\n",
            "w'(0) = -7  ->  w(1)= 3.6442\n"
          ]
        },
        {
          "data": {
            "text/plain": [
              "<matplotlib.legend.Legend at 0x7ff0e4d85240>"
            ]
          },
          "execution_count": 19,
          "metadata": {},
          "output_type": "execute_result"
        },
        {
          "data": {
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jdMI//iP867/Ctdfa3Rr7+HVJoyt5X2aM6Qm8ClzS1mLtrT9jxgzy8/MByM7O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\nokldKaXiiCZ1pZSKI/8/S//F/FKGxLYAAAAASUVORK5CYII=\n",
            "text/plain": [
              "<matplotlib.figure.Figure at 0x7ff0e5093240>"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "\n",
        "shoot(0)\n",
        "shoot(-5)\n",
        "shoot(-7)\n",
        "\n",
        "pt.grid()\n",
        "pt.legend(loc=\"best\")"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "See if you can find another solution to the boundary value problem by starting with $w'(0)=-30$.\n",
        "\n",
        "(You may call `pt.legend` to take advantage of automatic labeling.)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 18,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "w'(0) = -30  ->  w(1)= -1.4668\n",
            "w'(0) = -33  ->  w(1)= -0.22938\n",
            "w'(0) = -36  ->  w(1)= 1.062\n"
          ]
        },
        {
          "data": {
            "text/plain": [
              "<matplotlib.legend.Legend at 0x7ff0e5138c50>"
            ]
          },
          "execution_count": 18,
          "metadata": {},
          "output_type": "execute_result"
        },
        {
          "data": {
            "image/png": 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bcDac9Z45zTMioqJ33njDooIdk/ZOwvukN7s677I4dDMxPn7t6rEWdetCgwbw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\n5amV9N/Un8PdD5P9leyxNjV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bap5AcLB6ETx5ol4WadPCqVNQrZp6KWTL9vJf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\nV2S88DvnXJHxwu+cc0Xm/ydo9+EXFTJOAAAAAElFTkSuQmCC\n",
            "text/plain": [
              "<matplotlib.figure.Figure at 0x7ff0e4deaf28>"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "\n",
        "shoot(-30)\n",
        "shoot(-33)\n",
        "shoot(-36)\n",
        "\n",
        "pt.grid()\n",
        "pt.legend(loc=\"best\")"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": []
    }
  ],
  "metadata": {
    "kernelspec": {
      "display_name": "Python 3",
      "language": "python",
      "name": "python3"
    },
    "language_info": {
      "codemirror_mode": {
        "name": "ipython",
        "version": 3
      },
      "file_extension": ".py",
      "mimetype": "text/x-python",
      "name": "python",
      "nbconvert_exporter": "python",
      "pygments_lexer": "ipython3",
      "version": "3.5.1+"
    }
  },
  "nbformat": 4,
  "nbformat_minor": 0
}